HOSPITAL WAITING LISTS

Long waits for medical treatment are often used to indict governmental policies on healthcare systems. But a new study indicates that a proportion of long delays may be an integral feature of any such system, regardless of whether provisions, such as more staff, are made available to alleviate the queueing. Dominic Smethurst and H. C. Williams of the University Hospital in Nottingham have looked at the statistics of waiting lists for four dermatology specialists to whom patients were referred by their family doctors. The researchers followed the month-to-month variations in the delay between referral and appointments over six years.

Unsurprisingly, these delays fluctuate, all sorts of factors can affect the number of patients being referred or the availability of the specialists. A reasonable first guess would be to suppose that these fluctuations are random. That would mean there would be some well-defined average delay, with a few fortunate patients incurring shorter-than-average waits and a few unfortunates having to wait longer than usual. But Smethurst and Williams found something quite different, there was no real average. Instead, the fluctuations seemed to be following a particular type of mathematical relationship known as a power law. In essence, this means that when the waiting time doubles, the chance of waiting that long decreases by a fixed amount.

This kind of behaviour is common. Earthquakes, for example, follow a power-law relationship between size and probability. So too do the fluctuations of prices, exchange rates or performance in economic markets. Power laws are generally an indication that variations are controlled by interactions between the various parts of the system, rather than being driven by unconnected, chance events. In the case of medical waiting lists, these interactions might arise in all sorts of ways. Patients might be put on one list, for example, when other lists are seen to be long. And specialists might try to apportion time or appointments depending on how busy they are.

Whatever the cause, power-law behaviour has important implications. It is generally an 'emergent' property of a system, often called self-organization, which does not depend on the details of individual cases or interactions. This means that it can't be eliminated by minor tweaks, for example, by adding another specialist or more resources. That is likely to cause the waiting lists to reorganize themselves into another power-law distribution, numerically different, but qualitatively the same. And it is one of the features of a power law that it incurs a disproportionately high number of large fluctuations, long waiting times, in this case, relative to a system with purely random fluctuations.

If health system waiting lists do behave this way, long delays may be hard to eliminate. But conclusions should not be drawn too rapidly from the present analysis. For one thing, its authors see the same behaviour in the private and public health sectors, suggesting that there is nothing inherently inefficient about the latter. Second, power laws are notoriously hard to identify with small data sets, and the present data may not yet be adequate to rule out other types of mathematical relationship. Finally, power-law behaviour is commonly observed in studies of queuing, it is seen, for example, in the time delays for Internet transmissions. If it does turn out to be a feature of hospital waiting lists, that will simply identify them as queues like many others.